Draw a rectangle around all the pertinent clips with the mouse or select all. Hit shift+T to add the default audio transition. If you want to change the duration of all transitions after performing this operation, deselect, then redraw the rectangle or select all, click one of the transitions and edit it in the Inspector.

hrmm, this close, but not exactly what I want. It seems that this does a cross fade. I want to silence the audio from the end of the clip for one frame. The only way I know how to do that is to drag the fade in from the end of the clip.

By definition, a transition can't operate on one frame. If that's what you really want, I think you'd need to use a key-framed volume control.

If you can live with a change over 2 frames or more, go into Trim mode ("t"), select all the clips, then move the transition into the clip "out" position with "U". Then Shift+t. If you're doing a lot of them, it would probably be easier to change the default transition to the number of desired frames, first.